Formulas

Borrow Formulas

interestPerYear to interestPerSecond

$i_\text{second} = (i_{year})^\frac{1}{31622400}$

with both $i_{year}$ and $i_{second}$ interest accrual factors (and not interest rates) and where it is assumed that one full year consists of 31622400 seconds (366 days with 86400 seconds each).

Find a sample calculation here.

interestPerSecond to interestPerYear

$i_\text{year} = (i_{second})^{31622400}$

with both $i_{year}$ and $i_{second}$ interest accrual factors (and not interest rates) and where it is assumed that one full year consists of 31622400 seconds (366 days with 86400 seconds each).

interestPerSecond to interestToMaturity

$i_{\text{maturity}} = \begin{cases} (i_{\text{second}})^{T-t} & \text{if } t < T \\ 1e^{18} &\text{else} \end{cases}$

where $t$ is the current block.timestamp and $T$ is the collateral asset’s maturityboth expressed in seconds.

normalDebt to debt

$d = \frac{d_n*\text{rate}}{1e^{18}}$

debt to normalDebt

$d_n = \begin{cases} \frac{d*1e^{18}}{\text{rate}} &\text{if}\quad \text{rate}>0 \\ \infty &\text{else} \end{cases}$

where it is assumed that $rate \geq 1e^{18}$.

The following correction has to be performed on the resulting amount $d_n$ to avoid rounding errors due to precision loss:

$d_n = \begin{cases} d_n+1 &\text{if } \frac{d_n*\text{rate}}{1e^{18}} < d \\ d_n &\text{else } \end{cases}$

normalDebt to debtAtMaturity

$d_m = d_n\frac{\text{rate} + i_{\text{maturity}} - 1^{18}}{1^{18}}$

collateralizationRatio

$r = \begin{cases} \frac{p*c}{d} &\text{if}\quad d>0 \\ \infty &\text{else} \end{cases}$

Max.debt for a given collateralizationRatio and collateral amount

$d_\text{max} = \begin{cases} \frac{p*c}{r} &\text{if}\quad r>0 \\ \infty &\text{else} \end{cases}$

Min. collateral for a given collateralizationRatio and debt amount

$c_\text{min} = \begin{cases} \frac{r*d}{p} &\text{if}\quad p>0 \\ \infty &\text{else} \end{cases}$

Leverage Formulas

Min. collateralizationRatio for a levered deposit

$r_{\text{min}}=\frac{\frac{p*x_{\text{f→u}}}{1e^{18}}x_{\text{u→c}}}{1e^{18}}$

where $x_{f→u}$ and $x_{u→c}$ are the FIAT/underlying and underlying/collateral exchange rates including price impact and slippage.

Max. collateralizationRatio for a levered deposit

$r_{\text{max}}=\begin{cases} \frac{p*(c + \frac{x_{\text{u→c}}u}{1e^{18}})}{d} &\text{if } d>0 \\ \infty &\text{else} \end{cases}$

where $u$ is the deposited underlier amount and $x_{uc}$ is the underlying/collateral exchange rate including price impact and slippage.

flashloan amount for a levered deposit

$f=\frac{p*(c + \frac{x_{\text{u→c}}u}{1e^{18}}) - rd}{1e^{18}r - \frac{px_{\text{f→u}}}{1e^{18}}x_{\text{u→c}}}$

where $u$ is the deposited underlier amount and $x_{f→u}$and $x_{u→c}$ are the FIAT/underlying and underlying/collateral exchange rates including price impact and slippage.

Min. collateralizationRatio for a levered withdrawal

$r_{\text{min}} = \begin{cases} \frac{p(c - \Delta c)}{d} &\text{if}\quad \Delta c < c \text{ AND } d>0 \\ \infty &\text{else} \end{cases}$

where $c$ and $d$ are the position collateral and debt, and $\Delta c$ is the withdrawn collateral amount.

Max. collateralizationRatio for a levered withdrawal

$r_{\text{max}} = \begin{cases} \frac{p(c-\Delta c)}{d - \Delta c x_{c\rightarrow u}x_{u\rightarrow f}} &\text{if}\quad \Delta c < c \text{ AND } d>0 \\ \infty &\text{else} \end{cases}$

where $c$ is the position collateral and $\Delta c$ the withdrawn collateral amount.

flashloan amount for a levered withdrawal

$f = \begin{cases} d-\frac{p(c - \Delta c)}{r} &\text{if}\quad c>\Delta c \\ d &\text{if else}\quad c=\Delta c \\ \text{revert} &\text{else}\end{cases}$

where $c$ and $d$ are the position collateral and debt, and $\Delta c$ is the withdrawn collateral amount. It is thus assumed that $\Delta c \leq c$ and $d>0$ as the computation would otherwise yield an invalid result.

Estimated underlier for a levered withdrawal

$w = \frac{(\Delta c - \frac{\frac{f*1e^{18}}{x_{u→f}}1e^{18}}{x_{c→u}})x_{c→u}}{1e^{18}}$

where $\Delta c$ is the withdrawn collateral amount, $f$ is the flashloan amount used for the withdrawal, and $x_{u→f}$ and $x_{c→u}$ are the underlying/FIAT and collateral/underlying exchange rates including price impact and slippage.

Note that for a collateral asset beyond its maturity the formula remains intact with the difference that the input underlying/collateral exchange rate is fixed, i.e. $x_{u→c} = 1e^{18}$.

profitAtMaturity for a levered deposit

$g = w - u$

where $w$ is the estimated underlier amount withdrawn at maturity (i.e. with an input of $x_{cu} = 1e^{18}$ and $\Delta c = c$) and $u$ is the deposited underlier amount. It is further assumed that the collateral token can be redeemed for underlier tokens at a rate of $x_{cu}=1e^{18}$.

yieldToMaturity for a levered deposit

$y_{\text{maturity}} = \begin{cases} \frac{(u + g)1e^{18}}{u} - 1e^{18} &\text{if}\quad u>0 \\ \infty &\text{else} \end{cases}$

where $u$ is the deposited underlier amount and $g$ is the estimated profitAtMaturity.

yieldToMaturity to annualYield

$y_{\text{year}} = \begin{cases} (1e^{18}+y_{\text{maturity}})^{\frac{31622400}{T-t}}-1e^{18} & \text{if } t < T \\ 0 &\text{else} \end{cases}$

Trade Formulas

Minimal amountOut for a max slippagePercentage

$a_s = \frac{a(1e^{18}-s_\text{percentage})}{1e^{18}}$

where $a$ is the estimated amountOut without accounting for slippage.